Adaptive FDR control under independence and dependence

In the context of multiple hypotheses testing, the proportion $\pi_0$ of true null hypotheses among the hypotheses to test is a quantity that often plays a crucial role, although it is generally unknown. In order to obtain more powerful procedures, recent research has focused on finding ways to estimate this proportion and incorporate it in a meaningful way in multiple testing procedures, leading to so-called "adaptive" procedures. In this paper we focus on the issue of False Discovery Rate (FDR) control and we present new adaptive multiple testing procedures with control of the FDR, respectively under independence, positive dependencies (PRDS) or unspecified dependencies between the $p$-values. First, we present a new "one-stage" adaptive procedure and a new ``two-stage'' adaptive procedure that control the FDR in the independent context. We also give a review of other existing adaptive procedures that have provably controlled FDR in this context, and report extensive experimental results comparing these procedures and testing their robustness when the independence assumption is violated. Secondly, we propose adaptive versions of step-up procedures that have provably controlled FDR under positive dependencies and unspecified dependencies of the $p$-values, respectively. These are to our knowledge among the first theoretically founded adaptive multiple testing procedures that control the FDR when the $p$-values are not independent.